4  Further Exercises

Use R code to answer each of the following exercises.

Exercise 1

A university offers 8 different statistics courses to its honours level students. The numbers of students in each course are as follows:

• There are three course which have 85 students enrolled
• There are two course which have 112 students enrolled
• There is one course with 108 students enrolled
• There is one course with 95 students enrolled
• There is one course with only 46 students enrolled

Suppose you were to randomly sample three of these statistics courses with replacement. What would the sampling distribution of the sample mean number of students enrolled, \(\bar X\), be? What would the sampling distribution of the sample variance in the number of students enrolled, \(S^2\), be?

Can you use your sampling distributions to find \(E[\bar X]\), \(\sigma_{\bar X}^2\) and \(E[S^2]\)?

Exercise 2

Now suppose you want to randomly sample three of the above statistics courses without replacement. Find the sampling distribution of the sample mean number of students, \(\bar X\) and the sampling distribution of the sample variance in the number of students enrolled, \(S^2\).

Use these sampling distributions to find \(E[\bar X]\), \(\sigma_{\bar X}^2\) and \(E[S^2]\).

How do these values compare to those found in Exercise 1?

Exercise 3

Draw a random sample of 100 values from the \(\mbox{Poisson}(3)\) distribution and save these in a vector x.

Use the vector x to find the probability that the a variable following the \(\mbox{Poisson}(3)\) distribution is less than or equal to each value in x. That is, find \(P(X\leq x)\) when \(X\sim\mbox{Poisson}(3)\) for each value in x. Save these probabilities in a new vector called p.

Create a data frame called poisson which has a column called “draw” which is equal to the vector x and a second column called ’probability” which is equal to the vector p.

Exercise 4

The volume of a cylinder is calculated as \(V_{\mbox{cylinder}}=\pi r^2h\) where \(r\) is the radius of the cylinder and \(h\) is the height.

1. Write a function called cyl.vol which takes the arguments r and h and returns the volume of cylinder with radius r and height h.

   Use your function to find the volume of a cylinder with which has radius 2.8cm and height 24cm.

2. Use your function cyl.vol to write another function which can be used to calculate the total volume of \(n\) cylinders of the same size. This second function should take the arguments r for the radius, h for the height and n for the number of cylinders.